Unit 29 Using Graphs to Solve Equations

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Unit 29Using Graphs to Solve Equations
Presentation 1 Solution of Simultaneous Equations Using Their Graphs
Presentation 2 Graphs of Quadratic Functions
Presentation 3 Graphs of Cubic Function
Presentation 4 Reciprocal Functions
Presentation 5 Graphical Solutions of Equations
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Unit 29

• 29.1 Solution of Simultaneous Equations Using
  Their Graphs

3
Example Solve the pair of simultaneous equations
by drawing their graphs Solution We can
rewrite each equation in the form y
..... Plotting the lines,
when when Plot these
points Intersection at
y
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8
7
6
5
4
3
2
1








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x
0 1 2 3 4 5 6 7 8 9 10 11
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Unit 29

• 29.2 Graphs of Quadratic Functions

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Quadratic functions contain an x² term as well as
multiples of x and a constant. The following
graphs show 3 examples.
y
y
x
y
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1






























y
0
0
0
-3 -2 1 1 2 3
-3 -2 1 1 2 3
x
-3 -2 1 1 2 3
x
x
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Discuss the shape of the examples below.
y
y
y
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1
8
7
6
5
4
3
2
1






























0
0
x
0
-3 -2 1 1 2 3
-3 -2 1 1 2 3
-3 -2 1 1 2 3
x
x
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Unit 29

• 29.3 Graphs of Cubic Function

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Cubic functions involve an x³ term and possibly
x², x and constant terms as well. The graphs
below show some examples

• The graph of a cubic function can
• Cross the x-axis once as in example (a)
• Touch the axis once as in example (b)
• Cross the x-axis three times as in example (c)

6
5
4
3
2
1

-1
-2
-3
-4
-5
-6
6
5
4
3
2
1

-1
-2
-3
-4
-5
-6
6
5
4
3
2
1

-1
-2
-3
-4
-5
-6
y
y












y
























-3 -2 -1 1 2 3
0
-3 -2 -1 1 2 3
0
-3 -2 -1 1 2 3
0
x
x
x
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Unit 29

• 29.4 Reciprocal Functions

10
Reciprocal functions have the form of a fraction
with x as the denominator. The graphs below show
some examples
The curves are split into two distinct parts. The
curves get closer and closer to the axis, as
. The curves have two lines of symmetry,
and .
y
y
y
6
5
4
3
2
1

-1
-2
-3
-4
-5
6
5
4
3
2
1

-1
-2
-3
-4
-5
6
5
4
3
2
1

-1
-2
-3
-4
-5




































-3 -2 -1 1 2 3
0
-3 -2 -1 1 2 3
0
-3 -2 -1 1 2 3
0
x
x
x
11
The curves are split into two distinct parts. The
curves get closer and closer to the axis, as
. The curves have two lines of symmetry,
and . Where on the grids
below would the stated curves lie.
y
y
6
4
2

-2
-4
-6
6
4
2

-2
-4
-6
























-6 -4 -2 2 4 6
-6 -4 -2 2 4 6
0
0
x
x
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Unit 29

• 29.5 Graphical Solutions of Equations

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Use the graph to determine
(e) The value of at which is a
minimum (f) The interval on the domain for
which is less than

• (a) The value of when
• (b) The value of when

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(c) The value of when or (d) The minimum
value of
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y

















8
6
4
2

-2
-4
-6
-8
-3 -2 -1 1 2 3 4 5
0
x
14

• (a) Given that , complete this table.
• (b) Graph this equation for

x -2 -1 0 2 4 6
y 30 15 4 -6 0 22
?
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(c) Use this graph to solve Draw the line
on the grid. This intersects at
or
30
25
20
15
10
5

-5
-2 -1 1 2 3 4 5 6